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- 1. 1 A PowerPoint™Tutorial to Accompany macroeconomics, 5th ed. N. Gregory Mankiw ® Economic Growth
- 2. 2 А2 А1 А0 А3 А3 А0 А1 А2 Хэрэглээний бүтээгдэхүүн
- 3. 3 The Solow Growth Model is designed to show how growth in the capital stock, growth in the labor force, and advances in technology interact in an economy, and how they affect a nation’s total output of goods and services. Let’s now examine how the model treats the accumulation of capital.
- 4. 4
- 5. 5 The production function represents the transformation of inputs (labor (L), capital (K), production technology) into outputs (final goods and services for a certain time period). The algebraic representation is: Y = F ( K , L ) The Production FunctionThe Production Function IncomeIncome isis some function ofsome function of our given inputsour given inputs Let’s analyze the supply and demand for goods, and see how much output is produced at any given time and how this output is allocated among alternative uses. Key Assumption: The Production Function has constant returns to scale. z zz
- 6. 6 This assumption lets us analyze all quantities relative to the size of the labor force. Set z = 1/L. Y/ L = F ( K / L , 1 ) OutputOutput Per workerPer worker isis some function ofsome function of the amount ofthe amount of capital per workercapital per worker Constant returns to scale imply that the size of the economy as measured by the number of workers does not affect the relationship between output per worker and capital per worker. So, from now on, let’s denote all quantities in per worker terms in lower case letters. Here is our production function: , where f(k)=F(k,1).y = f ( k )
- 7. 7 MPK = f (k + 1) – f (k) yy kk f(k) The production function shows how the amount of capital per worker k determines the amount of output per worker y=f(k). The slope of the production function is the marginal product of capital: if k increases by 1 unit, y increases by MPK units. 1 MPK
- 8. 8 consumptionconsumption per workerper worker dependsdepends onon savingssavings raterate (between 0 and 1)(between 0 and 1) OutputOutput per workerper worker consumptionconsumption per workerper worker investmentinvestment per workerper worker y = c + iy = c + i1) c = (1-c = (1-ss)y)yc = (1-c = (1-ss)y)y2) y = (1-y = (1-ss)y + i)y + iy = (1-y = (1-ss)y + i)y + i3) 4) i =i = ssyyi =i = ssyy Investment = savings. The rate of saving s is the fraction of output devoted to investment.
- 9. 9 Here are two forces that influence the capital stock: • Investment: expenditure on plant and equipment. • Depreciation: wearing out of old capital; causes capital stock to fall. Recall investment per worker i = s y. Let’s substitute the production function for y, we can express investment per worker as a function of the capital stock per worker: i = s f(k) This equation relates the existing stock of capital k to the accumulation of new capital i.
- 10. 10 Investment, s f(k) Output, f (k) c (per worker) i (per worker) y (per worker) The saving rate s determines the allocation of output between consumption and investment. For any level of k, output is f(k), investment is s f(k), and consumption is f(k) – sf(k). yy kk
- 11. 11 Impact of investment and depreciation on the capital stock: ∆k = i –δk Change in Capital Stock Investment Depreciation Remember investment equals savings so, it can be written: ∆k = s f(k)– δk δk kk δk Depreciation is therefore proportional to the capital stock.
- 12. 12 Investment and Depreciation Capital per worker, k i* = δk* k*k1 k2 At k*, investment equals depreciation and capital will not change over time. Below k*, investment exceeds depreciation, so the capital stock grows. Below k*, investment exceeds depreciation, so the capital stock grows. Investment, s f(k) Depreciation, δ k Above k*, depreciation exceeds investment, so the capital stock shrinks. Above k*, depreciation exceeds investment, so the capital stock shrinks.
- 13. 13 Investment and Depreciation Capital per worker, k i* = δk* k1* k2* Depreciation, δ k Investment, s1f(k) Investment, s2f(k) The Solow Model shows that if the saving rate is high, the economy will have a large capital stock and high level of output. If the saving rate is low, the economy will have a small capital stock and a low level of output. An increase in the saving rate causes the capital stock to grow to a new steady state. An increase in the saving rate causes the capital stock to grow to a new steady state.
- 14. 14 The steady-state value of k that maximizes consumption is called the Golden Rule Level of Capital. To find the steady-state consumption per worker, we begin with the national income accounts identity: and rearrange it as: c = y - i. This equation holds that consumption is output minus investment. Because we want to find steady-state consumption, we substitute steady-state values for output and investment. Steady-state output per worker is f (k*) where k* is the steady-state capital stock per worker. Furthermore, because the capital stock is not changing in the steady state, investment is equal to depreciation δk*. Substituting f (k*) for y and δ k* for i, we can write steady-state consumption per worker as c*= f (k*) - δ k*. y - c + i
- 15. 15 c*= f (k*) - δ k*. According to this equation, steady-state consumption is what’s left of steady-state output after paying for steady-state depreciation. It further shows that an increase in steady-state capital has two opposing effects on steady-state consumption. On the one hand, more capital means more output. On the other hand, more capital also means that more output must be used to replace capital that is wearing out. The economy’s output is used for consumption or investment. In the steady state, investment equals depreciation. Therefore, steady-state consumption is the difference between output f (k*) and depreciation δ k*. Steady-state consumption is maximized at the Golden Rule steady state. The Golden Rule capital stock is denoted k*gold, and the Golden Rule consumption is c*gold. δk kk δk Output, f(k) c *gold k*gold
- 16. 16 Let’s now derive a simple condition that characterizes the Golden Rule level of capital. Recall that the slope of the production function is the marginal product of capital MPK. The slope of the δk* line is δ. Because these two slopes are equal at k*gold, the Golden Rule can be described by the equation: MPK = δ. At the Golden Rule level of capital, the marginal product of capital equals the depreciation rate. Keep in mind that the economy does not automatically gravitate toward the Golden Rule steady state. If we want a particular steady-state capital stock, such as the Golden Rule, we need a particular saving rate to support it.
- 17. 17 The basic Solow model shows that capital accumulation, alone, cannot explain sustained economic growth: high rates of saving lead to high growth temporarily, but the economy eventually approaches a steady state in which capital and output are constant. To explain the sustained economic growth, we must expand the Solow model to incorporate the other two sources of economic growth. So, let’s add population growth to the model. We’ll assume that the population and labor force grow at a constant rate n.
- 18. 18 Like depreciation, population growth is one reason why the capital stock per worker shrinks. If n is the rate of population growth and δ is the rate of depreciation, then (δ + n)k is break-even investment, which is the amount necessary to keep constant the capital stock per worker k. Investment, break-even investment Capital per worker, k k* Break-even investment, (δ + n)k Investment, s f(k) For the economy to be in a steady state investment s f(k) must offset the effects of depreciation and population growth (δ + n)k. This is shown by the intersection of the two curves. An increase in the saving rate causes the capital stock to grow to a new steady state. For the economy to be in a steady state investment s f(k) must offset the effects of depreciation and population growth (δ + n)k. This is shown by the intersection of the two curves. An increase in the saving rate causes the capital stock to grow to a new steady state.
- 19. 19 Investment, break-even investment Capital per worker, k k*1 Investment, s f(k) (δ + n1)k An increase in the rate of population growth shifts the line representing population growth and depreciation upward. The new steady state has a lower level of capital per worker than the initial steady state. Thus, the Solow model predicts that economies with higher rates of population growth will have lower levels of capital per worker and therefore lower incomes. k*2 (δ + n2)k An increase in the rate of population growth from n1 to n2 reduces the steady-state capital stock from k*1 to k*2. An increase in the rate of population growth from n1 to n2 reduces the steady-state capital stock from k*1 to k*2.
- 20. 20 The change in the capital stock per worker is: ∆k = i –(δ+n)kThe change in the capital stock per worker is: ∆k = i –(δ+n)k Now, let’s substitute sf(k) for i: ∆k = sf(k) – (δ+n)k This equation shows how new investment, depreciation, and population growth influence the per-worker capital stock. New investment increases k, whereas depreciation and population growth decrease k. When we did not include the “n” variable in our simple version– we were assuming a special case in which the population growth was 0. Now, let’s substitute sf(k) for i: ∆k = sf(k) – (δ+n)k This equation shows how new investment, depreciation, and population growth influence the per-worker capital stock. New investment increases k, whereas depreciation and population growth decrease k. When we did not include the “n” variable in our simple version– we were assuming a special case in which the population growth was 0.
- 21. 21 In the steady-state, the positive effect of investment on the capital per worker just balances the negative effects of depreciation and population growth. Once the economy is in the steady state, investment has two purposes: 1) Some of it, (δk*), replaces the depreciated capital, 2) The rest, (nk*), provides new workers with the steady state amount of capital. Capital per worker, k k*k*' The Steady State Investment,s f(k) Break-even Investment,(δ + n) k Break-even investment,(δ + n') k An increase in the rate of growth of population will lower the level of output per worker. sf(k)
- 22. 22 • In the long run, an economy’s saving determines the size of k and thus y. • The higher the rate of saving, the higher the stock of capital and the higher the level of y. • An increase in the rate of saving causes a period of rapid growth, but eventually that growth slows as the new steady state is reached. Conclusion: although a high saving rate yields a high steady-state level of output, saving by itself cannot generate persistent economic growth. Conclusion: although a high saving rate yields a high steady-state level of output, saving by itself cannot generate persistent economic growth.
- 23. 23
- 24. 24 The Production Function is now written as: Y = F (K, L × E) The term L × E measures the number of effective workers. This takes into account the number of workers L and the efficiency of each worker E. Increases in E are like increases in L.
- 25. 25Capital per worker, k k* The Steady State Investment, sf(k) (δ + n + g)k Technological progress causes E to grow at the rate g, and L grows at rate n so the number of effective workers L × E is growing at rate n + g. Now, the change in the capital stock per worker is: ∆k = i –(δ+n +g)k, where i is equal to s f(k) Technological progress causes E to grow at the rate g, and L grows at rate n so the number of effective workers L × E is growing at rate n + g. Now, the change in the capital stock per worker is: ∆k = i –(δ+n +g)k, where i is equal to s f(k) Note: k = K/LE and y=Y/(L × Ε). So, y=f(k) is now different. Also, when the g term is added, gk is needed to provided capital to new “effective workers” created by technological progress. Note: k = K/LE and y=Y/(L × Ε). So, y=f(k) is now different. Also, when the g term is added, gk is needed to provided capital to new “effective workers” created by technological progress. sf(k)
- 26. 26 Labor-augmenting technological progress at rate g affects the Solow growth model in much the same way as did population growth at rate n. Now that k is defined as the amount of capital per effective worker, increases in the number of effective workers because of technological progress tend to decrease k. In the steady state, investment sf(k) exactly offsets the reductions in k because of depreciation, population growth, and technological progress.
- 27. 27 Capital per effective worker is constant in the steady state. y = f(k) output per effective worker is also constant. But the efficiency of each actual worker is growing at rate g. So, output per worker, (Y/L = y × E) also grows at rate g. Total output Y = y × (E × L) grows at rate n + g. Capital per effective worker is constant in the steady state. y = f(k) output per effective worker is also constant. But the efficiency of each actual worker is growing at rate g. So, output per worker, (Y/L = y × E) also grows at rate g. Total output Y = y × (E × L) grows at rate n + g.
- 28. 28 Steady-state consumption is maximized if MPK = δ + n + g, rearranging, MPK - δ = n + g. That is, at the Golden Rule level of capital, the net marginal product of capital, MPK - δ, equals the rate of growth of total output, n + g. Because actual economies experience both population growth and technological progress, we must use this criterion to evaluate whether they have more or less capital than at the Golden Rule steady state. The introduction of technological progress also modifies the criterion for the Golden Rule. The Golden Rule level of capital is now defined as the steady state that maximizes consumption per effective worker. So, we can show that steady-state consumption per effective worker is: c*= f (k*) - (δ + n + g) k*c*= f (k*) - (δ + n + g) k*
- 29. 29
- 30. 30 An important prediction of the neoclassical model is this: Among countries that have the same steady state, the convergence hypothesis should hold: poor countries should grow faster on average than rich countries.
- 31. 31 ? The Endogenous Growth Theory rejects Solow’s basic assumption of exogenous technological change.
- 32. 32 Start with a simple production function: Y = AK, where Y is output, K is the capital stock, and A is a constant measuring the amount of output produced for each unit of capital (noticing this production function does not have diminishing returns to capital). One extra unit of capital produces A extra units of output regardless of how much capital there is. This absence of diminishing returns to capital is the key difference between this endogenous growth model and the Solow model. Let’s describe capital accumulation with an equation similar to those we’ve been using: ∆K = sY - δK. This equation states that the change in the capital stock (∆K) equals investment (sY) minus depreciation (δK). We combine this equation with the production function, do some rearranging, and we get: ∆Y/Y = ∆K/K = sA - δ
- 33. 33 ∆Y/Y = ∆K/K = sA - δ This equation shows what determines the growth rate of output ∆Y/Y. Notice that as long as sA > δ, the economy’s income grows forever, even without the assumption of exogenous technological progress. In the Solow model, saving leads to growth temporarily, but diminishing returns to capital eventually force the economy to approach a steady state in which growth depends only on exogenous technological progress. By contrast, in this endogenous growth model, saving and investment can lead to persistent growth.
- 34. 34 Solow growth model Steady state Golden rule level of capital Efficiency of labor Labor-augmenting technological progress Endogenous growth theory

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